From the Examples of agglutinative languages section of the Wikipedia page on Agglutination:
Japanese is also an agglutinating language, adding information such as negation, passive voice, past tense, honorific degree and causality in the verb form. Common examples would be hatarakaseraretara (働かせられたら), which combines causative, passive or potential, and conditional conjugations to arrive at two meanings depending on context "if (subject) had been made to work..." and "if (subject) could make (object) work", and tabetakunakatta (食べたくなかった), which combines desire, negation, and past tense conjugations to mean "(subject) did not want to eat".
In short, Japanese is considered an agglutinative language because what other languages express with helping verbs and other additional words (was not red, could not be made to eat), Japanese expresses with suffixes that are added to the word root (赤くなかった, 食べさせられなかった).
Nouns are usually not counted as agglutinative because they are conjugated with the copula and not with a suffix, but if one considers particles to be noun suffixes and not separate words, one could make the case that nouns are also agglutinative, especially since some particles can be agglutinated (私にも, 僕だけには).